Let S = {1, 2, 3, ...}, A relation R on S × S is defined by xRy if log_a x > log_a y when a \(\rm = \frac 1 2.\) Then the relation is:

Concept:

Relation Based on Logarithmic Comparison:

• A relation R is defined on the set S × S by xRy if log_ax > log_ay.
• The base of the logarithm is a = 1/2, which is a value between 0 and 1.
• For 0 < a < 1, the logarithmic function log_ax is a strictly decreasing function.
• Important property: If log_ax > log_ay, then it implies x < y when 0 < a < 1.
• Hence, the given relation reduces to xRy ⇔ x < y.
• Relation properties:
  • Reflexive: A relation is reflexive if xRx for all x in S. But x < x is never true ⇒ Not reflexive.
  • Symmetric: If x < y, then y < x is false ⇒ Not symmetric.
  • Transitive: If x < y and y < z ⇒ x < z ⇒ Transitive.

Calculation:

Given,

Relation: xRy ⇔ log_ax > log_ay

Base of logarithm: a = 1/2

⇒ Since 0 < a < 1, log_ax is a decreasing function

⇒ log_ax > log_ay ⇔ x < y

⇒ So, xRy ⇔ x < y

⇒ Check reflexivity: x < x ⇒ false ⇒ Not reflexive

⇒ Check symmetry: If x < y, then y < x ⇒ false ⇒ Not symmetric

⇒ Check transitivity: If x < y and y < z ⇒ x < z ⇒ Transitive

∴ The relation is transitive only.